The Penguins took a 2-1 lead at 4:23 of the first period when Anthony Mantha scored on a breakway.
Malkin gave Pittsburgh a 3-1 lead at 13:00 of the opening frame with a shot from the doorstep.
At 13:43 of the first period, Elmer Soderblom made it 4-1 with a shot from the left doorstep.
Burns cut the Avs' deficit to two at 14:21 of the first period with his 10th goal of the season via a one-timer from the right point.
With that goal, Burns recorded the 271st tally of his career and passed Bobby Orr for the seventh-most goals by a defenseman in NHL history.